SQLite does not compete with client/server databases.
SQLite competes with fopen().

Situations Where SQLite Works Well

Embedded devices and the internet of things

Because an SQLite database requires no administration,
it works well in devices that must operate without expert human support.
SQLite is a good fit for use in
cellphones, set-top boxes, televisions, game consoles,
cameras, watches, kitchen appliances, thermostats, automobiles,
machine tools, airplanes, remote sensors, drones, medical devices,
and robots: the "internet of things".

Client/server database engines are designed to live inside a
lovingly-attended datacenter at the core of the network.
SQLite works there too, but SQLite also thrives at the edge of the network,
fending for itself while providing fast and
reliable data services to applications that would otherwise
have dodgy connectivity.

Application file format

SQLite is often used as the on-disk file format
for desktop applications such as version control systems,
financial analysis tools, media cataloging and editing suites, CAD
packages, record keeping programs, and so forth. The traditional
File/Open operation calls sqlite3_open() to attach to the database
file. Updates happen automatically as application content is revised
so the File/Save menu option becomes superfluous. The File/Save_As
menu option can be implemented using the backup API.

There are many benefits to this approach, including improved
application performance, reduced cost and complexity, and
improved reliability. See technical notes
here and
here for details.

Websites

SQLite works great as the database engine for most low to
medium traffic websites (which is to say, most websites).
The amount of web traffic that SQLite can handle depends
on how heavily the website uses its database. Generally
speaking, any site that gets fewer than 100K hits/day should work
fine with SQLite.
The 100K hits/day figure is a conservative estimate, not a
hard upper bound.
SQLite has been demonstrated to work with 10 times that amount
of traffic.

The SQLite website (https://www.sqlite.org/) uses SQLite itself,
of course, and as of this writing (2015) it handles about 400K to 500K
HTTP requests per day, about 15-20% of which are dynamic pages touching
the database. Dynamic content uses about 200 SQL statements per webpage.
This setup runs on a single VM that shares a physical server with 23 others
and yet still keeps the load average below 0.1 most of the time.

Data analysis

People who understand SQL can employ the
sqlite3 command-line shell (or various third-party
SQLite access programs) to analyze large
datasets. Raw data can be imported from CSV files, then that
data can be sliced and diced to generate a myriad of summary
reports. More complex analysis can be done using simple scripts written
in Tcl or Python (both of which come with SQLite built-in) or in R or
other languages using readily available adaptors.
Possible uses include website log analysis, sports
statistics analysis, compilation of programming metrics, and
analysis of experimental results. Many bioinformatics researchers
use SQLite in this way.

The same thing can be done with an enterprise client/server
database, of course. The advantage of SQLite is
that it is easier to install and use and the resulting database
is a single file that can be written to a USB memory stick
or emailed to a colleague.

Cache for enterprise data

Many applications use SQLite as a cache of relevant content from
an enterprise RDBMS.
This reduces latency, since most queries now occur against the local
cache and avoid a network round-trip. It also reduces the load
on the network and on the central database server. And in many cases,
it means that the client-side application can continue operating during
network outages.

Server-side database

Systems designers
report success using SQLite as a data store on server applications
running in the datacenter, or in other words, using SQLite as the underlying
storage engine for an application-specific database server.

With this pattern, the overall system is still client/server:
clients send requests to the server and get back replies over the network.
But instead of sending generic SQL and getting back raw table content,
the client requests and server responses are high-level and
application-specific.
The server translates requests into multiple SQL queries, gathers the
results, does post-processing, filtering, and analysis, then constructs
a high-level reply containing only the essential information.

Developers report that SQLite is often faster than a client/server
SQL database engine in this scenario.
Database requests are serialized by the server, so concurrency is not
an issue. Concurrency is also improved by "database sharding":
using separate database files for different subdomains. For
example, the server might have a separate SQLite database for each
user, so that the server can handle hundreds or thousands of simultaneous
connections, but each SQLite database is only used by one connection.

File archives

The SQLite Archiver project shows how
SQLite can be used as a substitute for ZIP archives or Tarballs.
An archive of files stored in SQLite is only very slightly larger, and
in some cases actually smaller, than the equivalent ZIP archive.
And an SQLite archive features incremental and atomic updating
and the ability to store much richer metadata.

SQLite archives are useful as the distribution format for software
or content updates that are broadcast to many clients. Variations
on this idea are used, for example, to transmit TV programming guides
to set-top boxes and to send over-the-air updates to vehicle navigation
systems.

Replacement for ad hoc disk files

Many programs use
fopen(),
fread(), and
fwrite() to create and
manage files of data in home-grown formats. SQLite works
particularly well as a
replacement for these ad hoc data files.

Internal or temporary databases

For programs that have a lot of data that must be sifted and sorted
in diverse ways, it is often easier and quicker to load the data into
an in-memory SQLite database and use queries with joins and ORDER BY
clauses to extract the data in the form and order needed rather than
to try to code the same operations manually.
Using an SQL database internally in this way also gives the program
greater flexibility since new columns and indices can be added without
having to recode every query.

Stand-in for an enterprise database during demos or testing

Client applications typically use a generic database interface that allows
connections to various SQL database engines. It makes good sense to
include SQLite in the mix of supported databases and to statically
link the SQLite engine in with the client. That way the client program
can be used standalone with an SQLite data file for testing or for
demonstrations.

Education and Training

Because it is simple to setup and use (installation is trivial: just
copy the sqlite3 or sqlite3.exe executable to the target machine
and run it) SQLite makes a good database engine for use in teaching SQL.
Students can easily create as many databases as they like and can
email databases to the instructor for comments or grading. For more
advanced students who are interested in studying how an RDBMS is
implemented, the modular and well-commented and documented SQLite code
can serve as a good basis.

Experimental SQL language extensions

The simple, modular design of SQLite makes it a good platform for
prototyping new, experimental database language features or ideas.

Situations Where A Client/Server RDBMS May Work Better

Client/Server Applications

If there are many client programs sending SQL to the same
database over a network, then use a client/server database
engine instead of SQLite. SQLite will work over a network filesystem,
but because of the latency associated with most network filesystems,
performance will not be great. Also, file locking logic is buggy in
many network filesystem implementations (on both Unix and Windows).
If file locking does not work correctly,
two or more clients might try to modify the
same part of the same database at the same time, resulting in
corruption. Because this problem results from bugs in
the underlying filesystem implementation, there is nothing SQLite
can do to prevent it.

A good rule of thumb is to avoid using SQLite
in situations where the same database will be accessed directly
(without an intervening application server) and simultaneously
from many computers over a network.

High-volume Websites

SQLite will normally work fine as the database backend to a website.
But if the website is write-intensive or is so busy that it requires
multiple servers, then consider using an enterprise-class client/server
database engine instead of SQLite.

Very large datasets

An SQLite database is limited in size to 140 terabytes
(247 bytes, 128 tibibytes).
And even if it could handle larger databases, SQLite stores the entire
database in a single disk file and many filesystems limit the maximum
size of files to something less than this. So if you are contemplating
databases of this magnitude, you would do well to consider using a
client/server database engine that spreads its content across multiple
disk files, and perhaps across multiple volumes.

High Concurrency

SQLite supports an unlimited number of simultaneous readers, but it
will only allow one writer at any instant in time.
For many situations, this is not a problem. Writer queue up. Each application
does its database work quickly and moves on, and no lock lasts for more
than a few dozen milliseconds. But there are some applications that require
more concurrency, and those applications may need to seek a different
solution.

Checklist For Choosing The Right Database Engine

Is the data separated from the application by a network?
→ choose client/server

Relational database engines act as bandwidth-reducing data filters.
So it is best to keep the database engine and the data on
the same physical device so that the high-bandwidth engine-to-disk
link does not have to traverse the network, only the lower-bandwidth
application-to-engine link.

But SQLite is built into the application. So if the data is on a
separate device from the application, it is required that the higher
bandwidth engine-to-disk link be across the network. This works, but
it is suboptimal. Hence, it is usually better to select a client/server
database engine when the data is on a separate device from the
application.

Nota Bene:
In this rule, "application" means the code that issues SQL statements.
If the "application" is an application server and
if the content resides on the same physical machine as the application server,
then SQLite might still be appropriate even though the end user is
another network hop away.

Many concurrent writers? → choose client/server

If many threads and/or processes need to write the
database at the same instant (and they cannot queue up and take turns)
then it is best to select a database engine that supports that
capability, which always means a client/server database engine.

SQLite only supports one writer at a time per database file.
But in most cases, a write transaction only takes milliseconds and
so multiple writers can simply take turns. SQLite will handle
more write concurrency that many people suspect. Nevertheless,
client/server database systems, because they have a long-running
server process at hand to coordinate access, can usually handle
far more write concurrency than SQLite ever will.

Big data? → choose client/server

If your data will grow to a size that you are uncomfortable
or unable to fit into a single disk file, then you should select
a solution other than SQLite. SQLite supports databases up to
140 terabytes in size, assuming you can find a disk drive and filesystem
that will support 140-terabyte files. Even so, when the size of the
content looks like it might creep into the terabyte range, it would
be good to consider a centralized client/server database.

Otherwise → choose SQLite!

For device-local storage with low writer concurrency and less than a
terabyte of content, SQLite is almost always a better solution. SQLite
is fast and reliable and it requires no configuration or maintenance.
It keeps thing simple. SQLite "just works".